Glossary

This glossary defines key terms used throughout SPD Learn documentation.

BCI

Brain-Computer Interface. A system that translates brain activity (typically EEG) into commands for external devices. Motor imagery classification is a common BCI paradigm.

BiMap

Bilinear Mapping Layer. A neural network layer that performs the transformation \(Y = W^\top X W\) where \(W\) is constrained to the Stiefel manifold. This is the SPD analog of a linear layer.

Covariance Matrix

A matrix capturing the pairwise statistical relationships between variables: \(\Sigma_{ij} = \text{Cov}(X_i, X_j)\). Covariance matrices are symmetric and positive semi-definite (SPD if full rank).

CSP

Common Spatial Patterns. A spatial filtering technique for EEG that maximizes variance differences between classes. TensorCSPNet learns CSP-like filters through BiMap layers.

Domain Adaptation

Techniques for adapting a model trained on one domain (e.g., one recording session) to perform well on a different domain (e.g., another session) [Zanini et al., 2017]. TSMNet with SPDBatchNormMeanVar enables source-free unsupervised domain adaptation.

EEGSPDNet

A model using channel-specific (grouped) convolutions before covariance computation, allowing each EEG channel to learn independent temporal filters.

Eigendecomposition

Factorization of a matrix as \(X = U \Lambda U^\top\) where \(U\) contains eigenvectors and \(\Lambda\) is diagonal with eigenvalues. Fundamental operation for SPD matrix computations.

ExpEig

Exponential Eigenvalue Layer. The inverse of LogEig, mapping symmetric matrices back to the SPD manifold via matrix exponential: \(\expeig(X) = U \exp(\Lambda) U^\top\).

Filter Bank

Decomposition of a signal into multiple frequency bands using bandpass filters. TensorCSPNet processes filter bank representations to capture frequency-specific patterns [Ju and Guan, 2023].

Fréchet Mean

The generalization of the arithmetic mean to Riemannian manifolds. For SPD matrices, it minimizes the sum of squared geodesic distances. Used in SPDBatchNormMean and SPDBatchNormMeanVar for centering.

Geodesic

The shortest path between two points on a manifold. On the SPD manifold, geodesics are curves that preserve the Riemannian structure. The geodesic distance measures length along these paths.

Geodesic Distance

The length of the shortest path (geodesic) between two points on a manifold. For SPD matrices \(A\) and \(B\), the affine-invariant Riemannian distance is \(\frob{\log(A^{-1/2} B A^{-1/2})}\).

GREEN

Gabor Riemann EEGNet. A lightweight model using learnable Gabor wavelets for time-frequency feature extraction combined with SPDNet layers for classification.

Loewner Matrix

A matrix used in computing gradients through eigenvalue functions. For function \(f\), element \(L_{ij} = (f(\lambda_i) - f(\lambda_j)) / (\lambda_i - \lambda_j)\) when \(i \neq j\), and \(L_{ii} = f'(\lambda_i)\).

Log-Euclidean Distance

A computationally efficient distance metric for SPD matrices: \(\frob{\log(A) - \log(B)}\). Faster than geodesic distance but less geometrically accurate.

LogEig

Logarithmic Eigenvalue Layer. Maps SPD matrices to the tangent space via matrix logarithm: \(\logeig(X) = U \log(\Lambda) U^\top\). Converts the curved SPD manifold to flat Euclidean space for classification.

MAtt

Manifold Attention Network. A model that applies attention mechanisms on the SPD manifold, computing attention weights based on Log-Euclidean distance between SPD matrices.

Motor Imagery

Mental rehearsal of a motor action without physical execution. Produces characteristic EEG patterns (mu/beta desynchronization) that can be decoded for BCI applications.

Phase-Space Embedding

Reconstruction of a dynamical system’s state space from a single time series using time-delayed copies: \([x(t), x(t-\tau), x(t-2\tau), \ldots]\). Based on Takens’ embedding theorem.

PhaseSPDNet

A model that applies phase-space embedding (time-delay coordinates) to input signals before SPDNet processing, capturing nonlinear dynamical structure.

ReEig

Rectified Eigenvalue Layer. A non-linearity for SPD matrices that clamps eigenvalues to be above a threshold: \(\reeig(X) = U \max(\Lambda, \varepsilon) U^\top\). Similar to ReLU in standard networks.

Riemannian Manifold

A smooth manifold equipped with a Riemannian metric that allows measuring distances and angles. The space of SPD matrices forms a Riemannian manifold where standard Euclidean operations do not apply directly.

Sample Covariance

An estimator of the covariance matrix from samples: \(\hat{\Sigma} = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(x_i - \bar{x})^\top\). May not be SPD with few samples; use regularization.

SFUDA

Source-Free Unsupervised Domain Adaptation. Domain adaptation without access to source domain data during adaptation. Achieved in SPD Learn by updating SPDBatchNormMeanVar running statistics on unlabeled target data.

Shrinkage

A regularization technique that interpolates between the sample covariance and a structured estimate (e.g., diagonal): \(\hat{\Sigma}_{\text{shrink}} = (1-\alpha)\hat{\Sigma} + \alpha \cdot \text{target}\). Ledoit-Wolf shrinkage automatically selects optimal \(\alpha\).

SPD Matrix

Symmetric Positive Definite Matrix. A square matrix \(X\) that is symmetric (\(X = X^\top\)) and has all positive eigenvalues. SPD matrices form a Riemannian manifold and commonly arise as covariance matrices.

SPDBatchNormMean

Riemannian batch normalization that centers SPD matrices at their Fréchet mean: \(\tilde{P}_i = G^{-1/2} P_i G^{-1/2}\). Based on Brooks et al. (2019).

SPDBatchNormMeanVar

Extended batch normalization for SPD matrices that normalizes both mean (via Fréchet mean) and dispersion (via power transformation). Enables domain adaptation through domain-specific statistics.

Stiefel Manifold

The manifold of orthonormal matrices \(W\) satisfying \(W^\top W = I\). The weight matrices in BiMap are constrained to this manifold to preserve the SPD structure.

Tangent Space

The vector space of all possible directions at a point on a manifold. For SPD matrices, the tangent space at a point is the space of symmetric matrices. The LogEig operation projects SPD matrices to the tangent space at the identity.

TensorCSPNet

A model that stacks covariance matrices from multiple frequency bands into a tensor structure and processes them through SPDNet layers to capture temporal, spectral, and spatial information jointly.

Trivialization

A technique for optimizing manifold-constrained parameters by mapping from unconstrained Euclidean space to the manifold. SPD Learn uses trivialization for Stiefel manifold (BiMap weights) and SPD (batch norm parameters) constraints.

TSMNet

Tangent Space Mapping Network. A model combining convolutional features, covariance pooling, and SPDBatchNormMeanVar for domain adaptation in EEG classification.